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Theorem bnj132 32056
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj132.1 (𝜑 ↔ ∃𝑥(𝜓𝜒))
Assertion
Ref Expression
bnj132 (𝜑 ↔ (𝜓 → ∃𝑥𝜒))
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)

Proof of Theorem bnj132
StepHypRef Expression
1 bnj132.1 . 2 (𝜑 ↔ ∃𝑥(𝜓𝜒))
2 19.37v 1999 . 2 (∃𝑥(𝜓𝜒) ↔ (𝜓 → ∃𝑥𝜒))
31, 2bitri 278 1 (𝜑 ↔ (𝜓 → ∃𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  bnj996  32288
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